Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,15,12)(11,18,19,24)(13,17,14,22)(16,23,21,20);(6,7)(9,11)(10,16)(12,21)(13,20)(14,23)(15,19)(17,24)(18,22);(1,2,3,4)(5,6,8,7)(9,12,15,10)(11,16,19,21)(13,17,14,22)(18,23,24,20);(5,7,8,6)(11,19)(16,21)(18,24)(20,23);(9,13,15,14)(10,17,12,22)(11,20,19,23)(16,24,21,18)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72316431, #72316463 |
| maximal normal subgroup of | #72316431, #72316463 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372691);